No More Lower Arcs

 In the previous post "Looking For P4?" dated 02 Dec 2023, the arc $P1+P2+2$ is not excluded from providing a $q$.  This $q=1$, and $P3=P1+P2+2-2=P1+P2$.

And there is no need for full lower arcs, just long enough to provide the first integer marking.

This is how arc $P1+P2-2$ provides a possible solution always,

as $P1+P2+2$ is always even with $P1$ and $P2$ being prime and $P1+P2-2$ is short by four integer lengths.  Therefore, two marking (one on each portion) on arc $P1+P2-2$, will accommodate a line through it that intersect another integer marking on arc $P1+P2+2$.

Possibly,

$P4=P1-2$,     $P3=P2+2$

or

$P4=P1+2$,     $P3=P2-2$

when the radial intersects arc $P1+P2+2$ and recovers 4 integer units in total.